ADMS2320.com. We Make Stats Easy. Chapter 4. ADMS2320.com Tutorials Past Tests. Tutorial Length 1 Hour 45 Minutes

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1 We Make Stats Easy. Chapter 4 Tutorial Length 1 Hour 45 Minutes Tutorials Past Tests

2 Chapter 4 Page 1 Chapter 4 Note The following topics will be covered in this chapter: Measures of central location Measures of variability Percentiles Empirical Rule Linear Relationships How to use The goal of this website is to quickly and efficiently make you understand all the major concepts. We have many years of experience tutoring students for ADMS 2320 and we have a pretty good idea of what you need to concentrate on in order to do well. Each chapter, we will go through theory and then apply that theory through multiple choice and short answer questions. We attempt to make our videos quick and to the point. As such, it is a good idea to always review your class notes and textbook readings to go over additional topics and examples. Note: Not all theory is covered prior to answering practice questions. Some theory is taught while answering questions in order to put the theory into immediate practice. Make sure to go through all the videos in a given chapter.

3 Chapter 4 Page 2 Measures of Central Location - Mean Measures of Central Location In a given set of data, what value does the data cluster around? There are three types of measurements that can be used to measure central location: mean, median, and mode. Arithmetic Mean ("Average") Population Mean Sample Mean Legend - addition (sum) - the data point - the total data points in the sample - the total data points in the population A sample of five students have the following heights in inches: What is the mean height? (round to 4 decimal places) Excel generated the following numbers randomly and these numbers make up a population What is the mean of the randomly generated numbers? (round to 4 decimal places) The following table summarizes the number of hours a sample of five students studied and the test mark they received. What is the average test score? (round to 4 decimal places) Hours Studied Test Score (/100)

4 Chapter 4 Page 3 Measures of Central Location - Geometric Mean Geometric Mean Use the geometric mean when the variable represents growth rate or rate of change. Another way to look at it, is if we're trying to determine the average change over a period of time, use the geometric mean. - Geometric mean - Rate of return in year 1, 2, 3, etc A $50 investment grows by 100% in year one to $100. In year 2, there is a loss of 50% bringing the investment value down to $50. What is the arithmetic mean? Therefore average growth rate per year is 25% What is the geometric mean? Therefore average growth over 2 years is 0% A certain stock had the following yearly returns: Year Rate of Return Compute the mean and geometric mean (to four decimal places).

5 Measures of Central Location - Median and Mode Median The median is the middle data point. The median separates the lower 50% of the data and the upper 50% of the data. You must arrange the numbers in increasing or decreasing order and then find the number in the middle. If there are two numbers in the middle, the average of both must be taken to find the median. When data is arranged in order, the position of the median can be found using the following formula: Position of median = is the total number of data points. If the position is a whole number, that is the location of the median. If it is a decimal number, you have to average the data in the lower and upper position indicated by rounding the position down and then up. Mode The mode is the most frequently occurring observation in a dataset. It is possible to have more than one mode. In general, it helps to have the numbers arranged in increasing or decreasing order to see which number occurs the most often. Note: If every number in the data set occurs the same number of times, there is NO MODE. Find the median and mode of the following dataset Find the median and mode of the following dataset Find the median and mode of the following dataset Chapter 4 Page 4

6 Chapter 4 Page 5 Measures of Central Location Which method to use: mean, median, or mode? For interval data, any of them can be used, but usually it is the mean or median. If there are extreme observations in the dataset (some numbers way smaller or way larger), then usually the median is the better measure. The median is not affected by extreme values while the mean is. For ordinal data, use median. For nominal data, use mode Note: For nominal data, finding the mode doesn t technically give us a central location since it s not actually possible to find a center for nominal data However, by finding the mode, we found the most frequently occurring category. Shape of Distribution Based on Mean/Median/Mode Symmetric distribution: Mean = Mode = Median Positively Skewed Distribution: Mode < Median < Mean Negatively Skewed Distribution: Mode > Median >Mean

7 Measures of Variability Variability How much variation or spread is there between the data points in a given data set. Types of measures for variability: -Range -Variance -Standard Deviation -Mean Absolute Deviation (More will come as we go through the section) Range Range = Largest value Smallest Value Variance Population Variance Sample Variance Standard Deviation Population Standard Deviation Sample Standard Deviation Mean Absolute Deviation (MAD) Population MAD Sample MAD Chapter 4 Page 6

8 Chapter 4 Page 7 Measures of Variability Interpreting Standard Deviation and Variance Both numbers represent how much variability is in the data. The units for variance will be the original units squared, while the units for standard deviation are the units for the raw measurement. For example, if we were finding the variance of data collected on employee salaries, the variance would have units, but the standard deviation would have units of. Standard deviation tells us how far away a given data point is from the mean, on average. Coefficient Of Variation This is another measure of variability. This particular measure is unitless. This term allows us to see how much relative variability there is between datasets that have different units or different means. For example, a standard deviation of when the data set contains numbers in the s is not large. A standard deviation of 10 when the data set contains numbers less than 25 is large. Population coefficient of variation: Sample coefficient of variation:

9 Chapter 4 Page 8 Measures of Variability Find the range, variance, standard deviation, mean absolute deviation, and coefficient of variation for the following sample data points

10 Chapter 4 Page 9 Measures of Variability A physician has developed a new four week weight loss program. To determine if the program actually works, she records the starting weight and ending weight of all the participants. At the end of the program, she calculates the amount of weight lost. The weight loss for a random sample of eight participants is shown below Find the range, variance, standard deviation, mean absolute deviation (MAD), and coefficient of variation of the data set.

11 Chapter 4 Page 10 Percentile Percentile If a value X is at the more than X. Percentile, that means P% of the data is less than X and (100-P)% is For example, on a test out of 30, if a score of 25 is considered the 80th percentile, it means 80 percent of the scores are less than 25, and 20% of the scores are more than 25. The median is considered the 50th percentile because 50% of the data is below the median. The 25th percentile is called the first quartile (Q1) and the 75th percentile is called the third quartile (Q3). Interquartile Range (IQR) This is another measure of variability. IQR = Q3 - Q1 Locating Percentile Value The following formula indicates the position of the Pth percentile when the numbers are listed in increasing order. Location of th percentile: After determining the Value of th percentile = percentile, the following formula can be used to find the actual value: =The value located at when is rounded up to nearest whole number ("upper limit") =The value located at when is rounded down to to nearest whole number ("lower limit") = The decimal portion of

12 Chapter 4 Page 11 Percentile Consider the following dataset: A) What is the percentile? B) What is the third quartile? C) What is the IQR? D) What is the location of the 19th percentile? E) What is the 41st percentile?

13 Empirical Rule and Chebysheff's Theorem Empirical Rule The empirical rule only applies if the data distribution is bell-shaped 68% of data lies with 1 standard deviation of the mean 95% of data lies with 2 standard deviation of the mean 99.7% of data lies with 3 standard deviation of the mean,, or,, or or,, Chebysheff s Theorem For data that would not fit a bell-shaped distribution, the empirical rule cannot be used. Instead, we make use of Chebysheff's Theorem The minimum proportion of points that lie within for k > 1 standard deviations is : k=2 at least 3/4 (75%) of data lies within 2 standard deviations k=3 at least 8/9 (89%) of data lies within 3 standard deviations Chebysheff s Theorem gives us a lower bound on how much data lies within a given standard deviation. The empirical rule gives pretty accurate approximations of the total amount of data that will lie within a given standard deviation. Typically, any sort of skewed data would require the use of this theorem since it wouldn t be bell-shaped. Chapter 4 Page 12

14 Linear Relationships - Covariance When a scatterplot is drawn with 2 sets of interval data, there may be a positive or negative linear relationship that exists between the data. Two types of measurements that tell us about linear relationships are covariance and coefficient of correlation. Covariance Population Covariance N Sample Covariance Note: Sometimes the symbol is used to indicate covariance. Legend: X and Y represent the 2 different variables, and and are the corresponding data points. and represent the number of data point pairs in the sample or population Interpretation: If covariance is a large positive number, then as increases so does, or as decreases so does. If the covariance is a large negative number, then as increases decreases or vice versa. If the covariance is small, there is no general pattern between x and y large and small are relative terms that are hard to define To determine how strong the linear relationship is, we can use the coefficient of correlation. Chapter 4 Page 13

15 Chapter 4 Page 14 Linear Relationships - Coefficient of Correlation Coefficient of Correlation Population Coefficient of Correlation Sample Coefficient of Correlation The coefficient of correlation is essentially the covariance divided by product of the individual standard deviations of x and y. In general, the coefficient of correlation must be between -1 and 1. Strength Interpretation: 0 No linear relation weak linear relationship moderate linear relationship strong linear relationship 1 Perfect linear relationship Direction + for positive linear relation (as one variable increases, the other increases) - for negative linear relation (as one variable increases, the other decreases) Perfect negative correlation Perfect positive correlation 0.00 No correlation, i.e. no linear relationship between the variables

16 Chapter 4 Page 15 Linear Relationships Find the sample covariance and coefficient of correlation for the following sample data. Data 1 Data

17 Chapter 4 Page 16 Multiple Choice 1. Which of the following is not a measure of central location? a. mean b. median c. range d. mode 2. Which of the following is not always true regarding the mean? a. It is a measure of central location b. c. 50% of the data is above the mean, and 50% is below the mean. d. All of the above are always true regarding the mean 3. Consider the following dataset: What can you conclude regarding central tendency? a. The mean, median, and mode are equal b. The mean, median, and mode are unequal c. The mean and mode are equal d. The mode and median are equal 4. The is affected greatly by extreme values. a. mean b. median c. mode d. interquartile range 5. Find the sum of the squared deviations, if a sample has 12 observations and a variance of 5. a. 300 b. 55 c. 275 d For a skewed graph, how much data falls within three standard deviations? a. 75% b. 99.7% c. 89% d. Atleast 89% 7. The allows us to determine if a given standard deviation indicates great variability or little variability. a. median b. variance c. Coefficient of variation d. Correlation coeffcient.

18 Chapter 4 Page 17 Multiple Choice 8. If two variables are positively related (as one increases, so does the other), then the covariance will be: a. Close to 0 b. A large positive number c. A large negative number d. None of the above 9. If the coefficient of correlation is, this indicates: a. The slope is -1 b. The slope is 1 c. The slope is 0 d. All the observations fall on the line 10. Calculate the coefficient of correlation if a b c. 20 d. Not enough information to determine coefficient of correlation. 11. The percentile is always equal to the a. mean b. median c. mode d. All of the above